Theoretical and Mathematical Physics, Vol. 121, No. 1, 1999
INTEGRAL EQUATIONS FOR CORRELATION FUNCTIONS OF A Q U A N T U M O N E - D I M E N S I O N A L B O S E GAS N. A. Slavnov I The large-time, long-distance behavior of the temperature correlation functions of a quantum one-dimensional Bose gas is considered. W'e obtain integral equations, which are closely related to the thermodynamic Bethe ansatz equations and whose solutions describe asymptotic expressions. In the low-temperature limit, the solutions of these equations are expressed through observables of the model
1. Introduction This paper is a continuation of the study of correlation functions of a quantum one-dimensional Bose gas with delta-function interaction [1-5]. Here, we consider the problems related to the large-time, longdistance asymptotic behavior of the correlation functions. The main result of this paper is a system of integral equations (see Sec. 4), whose solutions are asymptotic expressions. The form of these equations is close to that of the thermodynamic Bethe ansatz equations, and their solutions seem closely related to the observables of the one-dimensional Bose gas model. Such a relationship exists in the low-temperature limit. The main object of our investigation is the temperature correlation function of local fields,
0)r
=
tr (e-H/Tq2(0, 0)~2~(X, t)) (1.1)
tr e - H / T
where T is temperature, H is the Hamiltonian
H : / dx (O~gdt(x)O~gd(x) + cCgt(x)gyt(x)g2(x)q(x) - hg2t(x)q(x)), c > 0 is the coupling constant, and h is the chemical potential. The operators r canonical Bose fields,
(1.2)
t) and g2t(x, t) are the
[@(x, t), q t ( y , t)] : 5(x - y).
(1.3)
The equation of motion corresponding to Hamiltonian (1.2) is called the quantum nonlinear SchrSdinger equation (quantum NLSE). For the reader's convenience, we here present the basic equations for observables of the one-dimensional Bose gas at a finite temperature. The energy of an elementary excitation (particle) satisfies the Yang-Yang equation [6]
z(A)=A 2-h-
~T / ?
K(A,p) l o g ( l + e_C(,)/T)dtL '
2c
K(A, p) - c2 + (,~ _ # ) 2
(1.4)
OO
I Steklov Mathematical Institute, RAS, Moscow, Russia, e-mail:
[email protected]. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 1, pp. 117-138, October, 1999. Original article submitted January 10, 1999. 1358
0040-5779/99/1211-1358522.00
(~) 1999 Kluwer Academic/Plenum Publishers
In the case with a positive chemical potential, the function e(,k) has the two real roots total spectral density of the vacancies in the gas is governed by the equation
2rrpt(A) = 1 +
F K(A,p)#(p)pt(p) d#,
e(+qT)
= 0. The
(i.5)
oo
where
(1.6)
b9(,~) = (1 + e ~ ( A ) / T ) - I
is the Fermi weight. The particle momentum, being a function of the spectral parameter, can also be found from the corresponding integral equation. However, only the derivative of the momentum with respect to the spectral parameter is needed below. This quantity coincides with the total density of the vacancies up to a coefficient,
ok(,x) (9 A
(1.7)
- 27rpt(A).
The particle speed v(,k) is 0e v(~)
-
Ok
k'()~)
27rpt (,k) "
(1.8)
The integral equation for the scattering phase is
2hE(A, v ) =
/_~o (3O
(ic + A - ~ ) K(A,#)tg(p)F(#,v)dp+ilogkic+ v .
(1.9)
Here and hereafter, the logarithm branch is -Tr < arglog z < 7r. The quantity S(A, v) = exp{27riF(A, u)} is the scattering matrix of two particles with the momenta k(A) and k(v). We use the thermodynamic observables above to describe the asymptotic expressions for the correlation function. Below, we consider the behavior of correlation function (1.1) as x ~ co, t --+ co with the fixed ratio )~o = x/2t. Because the one-dimensional Bose gas is massless, correlation function (1.1) has a powerlike decay at zero temperature, o
-Ao
The exponent Ao can be computed in the framework of conformal field theory [7]. At a finite temperature, the asymptotic expressions acquire an exponential decay,
(o2(0, O)~2)(x, t)) T ----+ Ct-~e-tlL
(1.10)
In other words, zero temperature is the phase transition point. T h e methods of conformal field theory can be applied at a finite temperature only approximately, for instance, for calculating the low-temperature contributions to the correlation radius [7]. The determinant representation m e t h o d [1-5] describes correlation functions at an arbitrary temperature and is especially powerful for studying their asymptotic behavior. In this paper, we consider the leading, exponential decay in Eq. (1.10). We consider neither the preexponential factor t -A nor the constant coefficient C in (1.10). In other words, our goal is to c o m p u t e the correlation radius. The paper is organized as follows. In Sec. 2, we briefly describe the method based on representing the correlation functions through Fredholm determinants and discuss the construction of the dual fields (the auxiliary quantum operators). In particular, we explain the reason for introducing these fields and 1359
their role in the correlation function calculations. (The details behind the review in Sec. 2 can be found in [1-5].) In Sec. 3, the methods for averaging the operators depending on dual fields are developed, and the "averaging mapping," which associates classical functions to the quantum operators, is defined. This mapping allows computing the asymptotic behavior of correlation function (1.1) (see Sec. 4 where the system of integral equations describing the asymptotic formulas is given). In Sec. 5, we consider some specific cases of correlation function (1.1). In particular, we prove that our results exactly coincide with the results in [7] in the low-temperature limit. 2.
Dual
fields
The method of the Fredholm determinant representation for evaluating correlation functions is described in detail, for example, in [8]. First, we must represent a correlation function in terms of a determinant of an integral operator, whose kernel depends on distance and time (and other physical parameters of the model). Such a representation was found for correlation function (1.1) in [t]. Further investigation of the obtained Fredholm determinant can be performed via the methods of classical exactly solvable equations because such determinants turn out to be r-functions of classical integrable systems. In particular, for the model of the q u a n t u m one-dimensional Bose gas, these determinants can be expressed through solutions of various classical NLSE (in the simplest case of the two-point field correlation function in the free-fermionic limit, the corresponding differential equation is the scalar NLSE). Solutions of classical integrable equations can be found using the Riemann-Hilbert problem approach. In particular, to calculate the large-time, long-distance asymptotic expressions, we can use the nonlinear steepest descent method [9], which allows obtaining the complete asymptotic expansion for correlation functions. Thus, our calculation of the asymptotic behavior of the correlation function proceeds in two stages: 1. representing the correlation function in the Fredholm determinant form and 2. calculating the asymptotic expressions for the obtained determinant using the methods of the Riemann-Hilbert problem and classical exactly solvable equations. However, this suffices only in the free-fermionic model case (in the quantum NLSE case, this corresponds to the limit of an infinite coupling constant: c ~ cx~). Outside the free-fermionic point, we must add one more step to the above scheme: 3. averaging with respect to an auxiliary vacuum. The present paper is devoted to this third stage of the calculations, which we consider in detail. To obtain the Fredholm determinant representation for the correlation functions of non-free fermionic models (a finite coupling constant for the one-dimensional Bose gas), we must introduce auxiliary quantum operators, the Korepin dual fields [10], because of the existence of a nontrivial S-matrix. Briefly, the form factor decomposition
(a159(x, t)o(o, O)la) = ~~'[(al59(0, O)lb)12eDob(~'t)
(2.1)
Ib) can be used to evaluate a correlation function of an operator (.9 with respect to a state [a). Here, the state Ib) ranges a complete set of states, and the function Dab(x, t) describes the dispersion law of the model. T h e quantity (alO(0 , 0)lb ) is called a form factor of the operator (_9; for example, in the quantum NLSE, it can be explicitly calculated via the algebraic Bethe ansatz. Thus, for 59 = g2, the form factor is proportional to a determinant, (alg2(0, 0)lb ) ~ det M.
(2.2)
We stress that in contrast to the quantum field theory models, the form factors here are constructed over the bare Fock vacuum, not over the physical vacuum. If the states ta) and Ib) are parameterized by the 1360
respective spectral parameters {~} and {#}, then the entries Mjk are functions of these spectral parameters,
= Mj,
{U}).
The character of the dependency of Mjk on the spectral parameters is very important. To obtain a determinant representation for mean (2.1), we must reduce the infinite sum of determinants in the r.h.s, of (2.1) to a single determinant. This can be done only if M j k = M(Aj, #k), i.e., the entries are parameterized by a single two-variable function M ( x , y) taken at the points x = Aj and y = #k- These are the so-called local matrices. Actually, the entries of the matrix Mjk, which describes form factor (2.2), contain the products of two-particle S-matrices ic + A - p ic + p - A'
S(A,#)-
(2.3)
which range all parameters {A} and {#}. This means t h a t the entries Mjk depend on all spectral parameters Mjk = M({A}, {p}) (nonlocal matrices) in the interacting fermion case. Therefore, it is impossible to reduce the sum of determinants (2.1) to a single determinant. On the other hand, in the free-fermionic limit (c --~ co), the S-matrix becomes trivial, and the entries Mjk hence depend only on two variables Mjk = M ( A j , #k). The details and all explicit equations can be found in [1]. Introducing the dual fields improves the situation. To reduce the matrix M to the local form, it suffices to factorize the S-matrix,
S(.X, U) = fl(s
(2.4)
where f l and f2 are some functions. Such a factorization is obviously impossible for S-matrix (2.3) if .fj are classical functions, but it can be easily performed in terms of quantum objects. We consider two creation operators q~(A) and qr and two annihilation operators pc(A) and pa(A) acting in an auxiliary Fock space as = (Olqo(
) = o,
=
(2.5)
= o.
The nonvanishing commutation relations are
[pr
= -[pr162
= log
~c+#
"
(2.6)
The dual fields are the linear combinations ~b(A) = q~(A) + p c ( A ) ,
r
= qr
+pc(A).
(2.7)
= 0,
(2.8)
Obviously, because of (2.6), the dual fields commute, [r
V(~)] = [r
r
= [r
r
but the vacuum expectation value of the expressions containing the dual fields can be nontrivial,
S(A,#) = - ic A - # _ zc+p-A
(O]er162
(2.9) 1361
Therefore, the S-matrix can be represented as the vacuum expectation value of the factorized expression of form (2.4). Because of property (2.8), the determinants depending on the dual fields are well defined. Therefore, the form factors can be represented in the form @O(0,
0)lb>
(2.10)
.~ det M = (01 det M(r r
where the matrix _84 is now local, i.e., Mjk = _si(Aj,pk). A similar representation can be obtained for the squared absolute value of form factor (2.1), and summation with respect to the complete set { Ib) } can be performed under the sign of the vacuum expectation value. Eventually, we obtain the representation for mean (2.1) in terms of the vacuum expectation value of a matrix determinant that depends on the dual fields in the auxiliary Fock space. In the thermodynamic limit, this matrix becomes an integral operator, which implies
(O(x, t)O(O, 0))
~ (01 det(I + V)IO),
(2.11)
where the kernel of the operator V depends on the dual fields r and r but because of (2.8), the Fredholm determinant is welI defined. We have described only the main idea of the method of dual fields. In actual cases, modifications are possible; for example, additional dual fields must sometimes be introduced (see [1] for details). However, the basic ingredient of the method is always factorization (2.4), which allows reducing the determinants of nonlocal matrices to local determinants. Therefore, the correlation function of a quantum one-dimensional Bose gas is proportional to the vacuum expectation value of the Fredholm determinant, which depends functionally on the dual fields. Nevertheless, the asymptotic analysis of this determinant can be performed as in the free-fermionic limit. Commutation relations (2.8) imply that at a certain stage of the calculations, the dual fields can be considered as classical functions analytical in the strip I~A l < c/2. This property follows from the representation of the dual fields in terms of the canonical Bose fields,
=
(2.12) =
+ c2/4
where [~1 (A),~v~(#)] = [~2(A),~2} (#)] = g(A - #),
~j(A)]0) = 0,
(0I~2}(A)= 0.
(2.13)
Commutation relations between the operators pr162and qe,r given by (2.12) exactly coincide with (2.6). We can find the large-time, long-distance behavior of Fredholm determinant (2.11) using the methods of the Riemann-Hilbert problem and classical integrable equations (see [2-5]). The asymptotic expression obtained is a functional of the dual fields treated as some classical functional parameters. To calculate the asymptotic behavior of the correlation function, we then exploit the quantum nature of these operators and average the obtained expression with respect to the auxiliary vacuum. The important point is that the averaging procedure and the procedure for calculating the asymptotic expressions do not commute with each other [5]. The asymptotic expression for the correlation function is equal to the asymptotic expression for vacuum expectation value (2.11), not to the vacuum expectation value of the asymptotic expression. This is why the results in [11] with regard to averaging expressions containing dual fields have a restricted application for evaluating correlation functions. Nevertheless, it 1362
was proved in [5] that there exist asymptotic representations that are stable with respect to the averaging procedure, i.e., the asymptotic expression for the vacuum expectation value is equal to the vacuum expectation value of the asymptotic expression. In what follows, we consider specifically such a representation [5] for the Fredholm determinant. 3.
The
averaging
map
T h e asymptotic formulas for the Fredholm determinant in terms of the dual fields [5] are rather cumbersome (see (4.1) and (4.2) below). Therefore, before averaging these expressions, we consider simpler, but rather general, examples. We use the technique developed here to average Eqs. (4.1) and (4.2) below. We consider two dual fields r and r defined by (2.7). We set the commutation relations between the creation and annihilation operators to be
=
(3.1)
=
where ~(A, p) is a two-variable function. We do not need the explicit form of this function here; we only note that by setting ~(A, #) = log(ic + p - ~) - log(ic + A - / ~ ) , we reproduce commutation relations (2.6). In addition to relations (3.1), we impose the constraint
(3.2)
[p~(~),q~(#)] = fl(A,#) = fl(#,A),
where r/(A, p) is a symmetrical two-variable function. We demonstrate below that additional constraint (3.2) does not change the final result, but it is convenient to set the function r/(~,#) nonzero for a time. T h e main property of dual fields (2.8) holds, but the vacuum expectation values can be nontrivial. We now evaluate the mean of the form
(01/:r162162
(3.3)
where F(~b, r and F(r are operator-valued functionals (we call them functionals for short) of the dual fields. T h e functional F does not depend on the field ~b, while ~- depends on this field linearly,
6r 2
=0.
(3.4)
All functionals (and functions) of the dual fields are understood as formal series. We begin with the formula (see [11])
det g'
(3.5)
where
E(r/)
=
exp [ ~
znzmrl(An,
A,-n)
.
(3.6)
n,r~=] 1363
In the 1.h.s. of (3.5), F ( r is a functional (or a flmction) of the field r a is a complex parameter, and the functions fk(z) are assumed to be holomorphic in a vicinity of the origin. In the r.h.s, of (3.5), the quantities zj are the roots of the system N
(3.7) If this system admits several solutions, then we must choose the unique solution that tends to zero as a --+ 0 (see [11]). Eventually, the Jacobian of system (3.7),
gjk = ~
Zj --o~fj
Zrn~(Am,#j)
,
(3.8)
is in the denominator of (3.5): It is convenient to modify result (3.5). For this, we introduce the function N
w(u) = ~ Zmd~m,U).
(3.9)
m-~-I
Multiplying each of Eqs. (3.7) by ~(,kj, u) and summing them all, we obtain N a(w) = ~(,.,) - ~ Z f-, ('~(,,,,)),%~,-,,,'-')
= o.
(3.10)
Thus, instead of system (3.7), we obtain a single equation (3.10) for the function w(u), which must hold for any value of u. The roots of the system zj can be expressed in terms of w(u) as
zj = c~fj (w(#j)).
(3.11)
It is easy to see that d e t g = det [rsa(~(~))
(3.12)
Result (3.5) then becomes
(01 exp
g ~ ar162
}
F(r
E(r/) = F ( w ( v ) ) det(SG/Sw)"
(3.13)
kml
We now introduce a dependence on the parameter t in the functional F(r F(r = F(r ). This parameter plays the role of time in the asymptotic formulas for the correlation function. Then, Eq. (3.13) can be written in the form N
ioj ox { Z. k=l
1364
J, ( (OIC+(r162215
r
+
~(A2) + "~"
-~-
] oo xexp{~-~ f_ (z-2At+ir
[e~(A)/T er ) sgn(A-]Q dA ] e,~)-Ty+-i w(A)] f [0) _
_
(4.2)
for h > 0. Equations (4.1) and (4.2) were obtained under the assumption that t --+ oc and x --+ oo with the fixed ratio x/2t = Ao. The function E(A) is the solution of Yang-Yang equation (1.4). The most important objects for us are the three dual fields r r and r The commutation relations between corresponding creation and annihilation operators are
=
[p~,(A),qr
= [/5r
=
-log
ic+
'
= [p,~(A),@(p)] = r/(A,#) -- log
( (A - ~-2" + c2 ) "
(4.3)
Customarily, all the dual fields commute with each other. This property allows considering these objects as classical functions at a certain stage of the calculations. We note that Eqs. (4.1) and (4.2) were obtained 1367
in [5] using this property of the dual fields, although some operator features were taken into account as well. In particular, the asymptotic formulas must be stable under the averaging procedure with respect to the auxiliary vacuum, i.e., the corrections, which depend on the dual fields, must remain small after averaging. This is why the value X =
+
(4.4)
(see the previous section) enters the asymptotic formulas. All expressions containing the operator A must be understood as series (3.22) and (3.23); in particular, the integrals depending on the sign function are equal by definition,
/_~f(A,sgn(~,-A))dA=/_~f(A,l)dA+~ f(A,-l)dA.
(4.5)
Formula (3.22) can be used for each of the integrals in the r.h.s, of (4.5). Actually, /~ results from the partial summation of the asymptotic series for the correlation function. Operators/~1 and A2 are treated analogously. If we consider the duaI fields as classical functions, then /~i are the roots of the equation
e(.}~i) = Tsgn(/~
-/~i)r
s
-'~ --qT, ilk2 --+ qT at T --+ 0,
(4.6)
where -{-qT are zeros of the Yang-Yang function, e(=t=qT) ---- 0. Thus, the numbers -~i correspond to zeros of the expression under the logarithm sign in integral (4.2). Actually, for the dual field r /~i become operators defined by a series similar to (3.23). We only demand that operator equation (4.6) becomes the usual equation under the averaging map E(Ai) = Tsgn(A - Ai)w(Ai),
A1 ~ --qT, A2 --+ qT at T --+ 0.
(4.7)
The function w(A), which enters (4.2), is w(A) = sgn(A -/~1) sgn(A - s These sign functions are defined the same as the sign functions of A, Eq. (4.5). Equations (4.1) and (4.2) also contain a function v(/~), which determines the power law of t,
(4.8)
The explicit form of the factors C•162162 is unknown (see [5]). However, they are certainly independent of the field r and depend on the distance x and time t only through the fixed ratio x/2t = A0. Although asymptotic formulas (4.1) and (4.2) look rather complicated, they belong to the class of functionals of the dual fields considered in the previous section. They are asymptotically linear with respect to the field r The operators Ai are some (implicit) functionals of the field r The presence of the third field r as well as the existence of the unknown factors C+, causes no trouble because the trivial commutation relations
1368
imply that the averaging of the field 0~ can only contribute to the common constant factor, which does not affect the correlation radius, and we can even set r = 0. Thus, applying averaging mapping (3.29) to Eqs. (4.1) and (4.2) and neglecting the common constant factor and powerlike dependence on t, we have (~2(O,O)~t(x,t))T ----+ e - ' l ' e
(4.9)
,
where r+ correspond to the respective cases of positive and negative chemical potential. We call r+ the correlation radii, although, strictly speaking, they are not necessarily real. For h < 0,
- 2it(A - ,\0)/3 + itA 2 - i x A - iht + r_
+ ~
(x - 2Xt) sgn(A - A)log
eT-(~)-~ +-i
(4.10)
d1,
where the function w(u) depending on the parameters A and/3 is the solution of the integral equation
w(u) = ~(A,u) + i/3h"(A,u)
1
-
~
/5
~o
K ( u , ~) sgn(A - ~) log [e~(X)/r - em(a)sg,(~-A) e~X)-TT +- i
t
d)~
(4.11)
with (ic+__)~~(A,p) = l o g \ i c + p
'
K(A,.)=-i
~(A,p) = ()~
#)" + d "
The parameters A and/7 are determined by the two additional equations i(A - .ko) + ~-~ ~-~
(~o - .k)sgn(A - A)log
e Z-(~)-7~Ti
ds = 0,
(4.13)
OQ
, oi_
i(/3 + A - Ao) + ~--~~-~
(A0 - A) sgn(A - A) log L
e ~T~)-7-~+- ]
j d1=0, (4.14)
where x = 2tA0 is taken into account. For the positive chemical potential, the formulas are similar; the correlation radius is it ix t _ 2it(A - ~o)/3 + ~ ( h ~ + A~ - 2h) - ~ ( A 1 + A~) + r+ + ~
oo(x - 2~t) sgn(A - ~)log [
e~-x)-77TT]-
w(X) d~,
(4.15)
w(,~) dA,
(4.16)
where the function w(u) satisfies the integral equation 1
w(u) = 7 (~(A1, u) + ~(A2, u)) + i/3h'(A, u) 1 27r
K(u,A)sgn(A-
,~)log/
1369
A1 and A2 are the roots of Eq. (4.7), and w(A) = sgn(A - A,) sgn(A - A2). The p a r a m e t e r s A and fl can be found from the additional equations i(A - Ao) + ~ ~
(Ao - A) sgn(A - A)log
e~-x)-~-+l
w(A) dA = O,
O~
1 0 ifl + 2~ 0-X
~
(4.17)
(Ao - A) sgn(A - A) log [ce()')/T -- ew(;q sgn(X-A)
;;-~V~+-i-
~(~)
d~ = 0.
(4.18)
The exponential decay of correlation function (1.1) is thus governed by the function w(u). This function can be found from a system of integral equations whose structure is similar to Yang-Yang equation (1.4). Therefore, the function w(u) can p r e s u m a b l y be expressed in terms of observables of the one-dimensional Bose gas, which is d e m o n s t r a t e d for some limit cases in the next section. To conclude this section, we consider the free-fermionic limit c --+ oo. Then, K(A, p) = ~(A, p) = 0 (see Eqs. (4.12)), and hence w(u) = 0. Moreover, E(A) = A2 - h, A = Ao,/3 = 0, and A1,; = : F v ~ . T h e integral term in Eqs. (4.10) and (4.15) becomes
le~(~)/T - 1[
l f_ ~
co Ix - 2At I log e~(~)/--------~i dA.
(4.19)
In the case with a positive chemical potential, this expression completely describes the a s y m p t o t i c behavior of the correlation function because A~ + A 2 - 2h = 0. For h < 0, we have the additional oscillating term - i t ( A 2 + h). These formulas exactly coincide with the results in [12]. T h e free-fermionic limit is not very representative, however. Indeed, in this case, all c o m m u t a t i o n relations (4.3) between the o p e r a t o r s p and q become trivial, and we can immediately set all the dual fields equal to zero in Eqs. (4.1) and (4.2). 5.
Limit
cases
In this section, we consider some limit cases of correlation function (1.1). The first case corresponds to the autocorrelation. Although we have considered the limit x -+ c~, t -+ co, the asymptotic analysis in [5] was essentially based on the a s s u m p t i o n t h a t the ratio x / 2 t = A0 remains finite. In particular, A0 can be set equal to zero, which corresponds to the case x = 0. For example, we consider the case h < 0. We can easily verify that we can set A = /3 = 0 in Eqs. (4.11)-(4.14) for A0 = 0. Indeed, Eq. (4.11) implies t h a t w(u) is an odd function for A = /3 = 0. Differentiating (4.11) with respect to A and /3 at the zero point, we find that the derivatives Ow/OA and Ow/O/3 are even functions of u. T h e n Eqs. (4.13) and (4.14) are satisfied automatically, and the result slightly simplifies, = ih - 2
/o
log,
(5.1)
j
where w ( u ) is the solution of the equation
w(u) = -~(u, 0) + 1 1370
(K(A,u) - K ( A , - u ) ) l o g [
e-;-~/~- + 1 j d A .
(5.2)
Similarly, for a positive chemical potential, we obtain
(5.3) and
w(u)
= ~1 (~(A,, u) - ~(A1, - u ) ) +
_ e~(~) w(A)] 1 fo ~ (K(A,u)-K(A,-u))log[ [ ee()')/r e-7~/~+l
dA.
(5.4)
Then, it turns out t h a t A2 = -A~. The most interesting is the low-temperature limit for the case h > 0. In this limit, we can compare our results with those previously obtained [7]. Indeed, at T = 0, the asymptotic formula for the correlation function must be powerlike. Therefore, the correlation radius r+ ~ oo. The correlation radius was calculated for a low t e m p e r a t u r e in the linear approximation in [7]. The low-temperature limit is therefore a good test for our results. Moreover, we can improve the corresponding estimates from [7]. We now present the system of equations describing the ground state for h > 0 [13]. First, it is YangYang equation (1.4) in which the function e(A) has two real roots, 6(+qT) : 0, ~(A) > 0 if IAt > qT, and e(A) < 0 if I,~1 < qr. At T --+ 0, the roots +qT tend to some fixed values (corresponding to the boundaries of the Fermi sphere), qT --+ q. It is easy to verify that
e_~(.>/~Lv2 ~ o, t ~,
[~1 > q, lul < q.
Thus, Eq. (1.4) becomes
1/,
co(A) = A2 - h + ~ - ~
q K(~,u)e0(u)du,
e0(+q) = 0.
(5.5)
(Here and hereafter, we label observables at zero temperature with the subscript 0.) Similarly, the equations for the density and the scattering phase have the respective forms
p0(~)= ~1+ ~ Fo(~,~)= ~1
1 f_~K(A,p)po(#)d#, q f ; q K(A,#)Fo(#,v)d#+ l~(A,u).
(5.6)
(5.7)
The derivatives of the momentum and the particle speed are
Oko(A) 0a
_
2~po(A),
v0(A) = r k;(a)
(5.8)
as before. The equation for the resolvent of the integral operator entering Eqs. (5.5)-(5.7) is (we recall that (I + R)(I - K / ( 2 r r ) ) = I)
(5.9) 1371
It is easy to see that R(A, p) = -OuFo(A,#). Furthermore, comparing Eqs. (5.6) and (5.7) and using the relation K(A, p) = - i 0 ~ ( A , p), we obtain 27rpo(A) = 1 + Fo(A,-q) - F0(A,q).
(5.10)
Similarly to the Yang-Yang equation, the integral equation for the function w(u) becomes linear in the limit T -+ 0. If, for example, Az < Ae < A, then we have 1
1
wo(u) = ~(~(-q,u) + ~(q,u)) + ij3K(u,A) + -~ f_~q K(u,A)wo(A)dA
(5.11)
at T = 0 instead of (4.16). Here, we take into account that A2 = -Aa = q for T = 0. Comparing Eq. (5.11) with (5.7) and (5.9), we obtain
wo(u) = -i~r(Fo(u,-q) + Fo(u,q)) + 27ci13R(u, A).
(5.12)
The numbers A and/3 are determined from the system i(A - Ao)
i3
1 0 2~r Off /_7q (A0 - A)wo(A) dA = 0,
1 0 27r OA f 7 q (Ao - A)wo(A) dA =
(5.13)
(5.14)
O.
It follows immediately from (5.12) and (5.14) that/3 = 0 and hence
wo(u) = -iTr(Fo(u, -q) + Vo(u, q)).
(5.15)
Substituting (5.15) in (4.15) (where the logarithm also becomes a linear function), we obtain
t _ it(q 2 _ h) +
r+
2
q
(x - 2At)(Fo(A,q) + Fo(A,-q)) dA.
(5.16)
The r.h.s, of (5.16) vanishes identically. Indeed, by virtue of (5.7), the integral in the r.h.s, of (5.16) can be represented in the form 1
f f f (x - 2M)(5(A - p) + R(A, I-t)) (~(U, q) + ~(P, -q)) dA dp. q
Acting by the resolvent on the left and using (5.5)-(5.8), we obtain
- -1
47r
' f ; q ( 9 ko(A )
-
q) +
-q))
Because k~(A) is an even function, the coefficient by x is equal to zero. After we integrate by parts, the remaining integral gives
47r j_q e~ 1372
q) + K ( A , - q ) ) dA.
Using Eq. (5.5) and the condition r
= 0, we obtain r+ 1 = 0
forT=0.
(5.17)
Therefore, our formulas produce the correct result at zero temperature. R e m a r k . We recall that introducing the dual fields is equivalent to factoring the S-matrix. It is natural to expect that after we average with respect to the dual fields, the objects pertaining to the scattering matrix must appear. However, the factorization occurs over the bare Fock vacuum, while the averaging is performed in the thermodynamic limit, i.e., over the physical vacuum. Then, instead of the "bare" S-matrix, the function Wo(U) arises, which can be expressed in terms of "dressed" scattering phase (5.7) at the boundary of the Fermi sphere. In the calculations above, we do not need the explicit value of A, which can nevertheless be easily found. It follows from Eq. (5.13) that
A - A0 -
f
(~o - s
A) ds = 0,
q
and we immediately obtain 1 !
s
- ~co(A) = 0
X
or
vo(A) = -.
t
(5.18)
Therefore, the physical sense of the quantity A is that a particle with the speed vo(A) passes the distance x during the time t. To compute the correlation radius for small, but nonzero, temperature, we need more accurate estimates of the integrals containing the logarithmic function. An example of such calculations is given in the appendix. However, it is clear that all low-temperature corrections to both the correlation radius and the integral equation for the function w(u) come from vicinities of the points 4-q defining the boundary of the Fermi sphere. The integral equation for the function w(u) arising in the first order in T is as follows. We seek the function w(u) as the series
w(u) = wo(u) + Twl(u) + T2w2(u) + . . . , where
Wo(U) is given by (5.15). For Wl(U), we have w l ( u ) - ~1
/_~q K(u,A)Wl()~)d)~- 47r~)(q) K(u,q) (wo(q) § iTr)2 K(u,-q) 47rc~(q) (w~
(5.19)
and hence T-2
wl (u) -- 2s;(q------~(1 -
Fo(q, q) - Fo(q, _q))2 (R(u, q) + R(u, -q)).
(5.20)
Here, we first take F o ( - A , - # ) = - F o ( A , p ) into account and second set the parameter /3 = 0. This is always true in the framework of the low-temperature approximation. Indeed, the dependence of w(u) on the sign functions sgn(A - u) disappears in all orders in T. The parameters A and/3 enter the equation for w(u) only in the combination/3K(A, u) (see (A.8)). The derivative cOw/OA is therefore always proportional to/3, and Eq. (4.18) is hence valid for/3 = 0. 1373
Substituting wl (u) in the answer for correlation radius (4.15), we obtain -1 7rT (1 - Fo(q, q) - Fo(q, -q))'- (Ix - vot] + Ix + vot[) r+ = 4v--o
(5.21)
after simple algebra, where vo = vo(q). This result is valid for arbitrary locations of the roots A1 and A: with respect to the point A. In the considered case, we have A1 < A2 < A and can drop the signs of the absolute value. Comparing (5.21) with the result obtained in [7],
_
~T [x-votl+lx~vot
r + l = 4vo
I
(2rpo(q))
(5.22)
we obtain the identity 1
27rpo(q)
= 1 - Fo(q,q) - F o ( q , - q ) .
(5.23)
Naively, such a nonlinear relationship between the solutions of the linear integral equations seems rather strange. However, taking the nonlinear identity for the scattering phase [14, 15] (1 - F o ( q , q ) ) 2 - F ~ ( q , - q ) =
1
(5.24)
and Eq. (5.10) into account, we reconstruct identity (5.23). Therefore, in the linear-in-temperature approximation, our result for the correlation radius exactly coincides with the result obtained in the framework of the conformal field theory. Calculating higher temperature corrections is standard. In the order T n, equations for wn(u) remain linear (see (A.8)), and their solutions can be written explicitly in terms of the resolvent of the operator I - (1/27r)K and its derivatives. Correspondingly, the correlation radius is expressed in terms of the functions k~(q), ~o(q), Fo(q, +q), and their derivatives. In particular, the correction of order T 2 to r+ 1 vanishes. Result (5.21) is therefore valid up to terms of order T 3. 6.
Conclusion
We have considered the large-time, long-distance asymptotic behavior of the correlation function of a quantum one-dimensional Bose gas. T h e asymptotic formulas can be expressed in terms of the solutions of integral equations, which are closely related to the equations of the thermodynamic Bethe ansatz. In the low-temperature limit, we have described the solutions of these equations in terms of the model observables. These integral equations arise when averaging with respect to the dual fields. In the earlier works (see, (,.g., [10]), these q u a n t u m operators were auxiliary and were only used to solve certain combinatorial problems; our results provide a new insight on these operators. In particular, the dependence of the solutions of the integral equations on the scattering phase at zero temperature is by no means accidental. Apparently, the role of the dual fields is much more fundamental than was understood before. Acknowledgments.
The author thanks A. R. Its and V. E. Korepin for the useful discussion.
Appendix As an example of the low-temperature expansion, we consider the integral in the Yang-Yang equation I = T
F
oo
1374
K(A, #) log(1 + e -e(~)/T) d#.
(A.1)
Obviously, at T = 0, only the interval I-q, q] contributes to this integral:
F
IT=O = --
q
K(A, #)e(#) dp,
e(+q) = 0.
(A.2)
At a low nonzero temperature, the corrections to this expression pertain to the contributions of vicinities of the points +q. For instance, we consider the integral
j~q OO
I+ = T
K(A, p)log(1 +
dp,
e
(A.3)
T
where e(qr) = O. Replacing the variables e(p) = Tz, we obtain
I+ = T 2
~o(A, Tz)log(l + e - : ) d z ,
(A.4)
~0 ~176
where ~o(A,Tz) = K(A, p ) / e ' ( # ) . Now expanding integral (A.4) in a power series in T, we obtain oo
I+ = ~
Tn+2~o(n)(A,0)(1 - 2-n-1)~(n + 2)
(A.5)
n-----0
or, in terms of the functions K(A, #) and E(#),
oo
I+ = ~ ] T " + 2 ( 1
. K(A, qr) e'(qT--~'
- 2 - " - a ) ~ ( n + 2)Dqr
(A.6)
n~0
where 1 0 D~ = E'(A----'~0--~"
All the remaining contributions from vicinities of the points +qT can be calculated similarly. We must then substitute the obtained expansion in the Yang-Yang equation and set oo
E(A) = ~ n----O
oo
Tn&~(A),
qT = q + ~
Tnqn.
(A.7)
n=l
The arising infinite set of equations can be solved recursively. As a result, the functions e , and parameters qn are expressed in terms of do(q) and the resolvent R(A, +q). Analogously, all integrals from Sec. 4 can be decomposed into series. We present the complete temperature expansion of integral equation (4.16) for the function w(u): -
1 f_'q K(u,A)w(k)dA = :~(~(-q,u)+~(q,u)) +il3K(A,u)1
-?=4-1 n----2
where B,~(z) are the Bernoulli polynomials. It is still necessary to substitute decompositions (A.7) into this equation, after which the functions w,~(u) can be expressed in terms of do(q), R(A, =t=q), and their derivatives. 1375
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